# Luennot 8.2.2007
########################
y<-8; n<-10
curve(dbinom(y,n,x),0,1)
y<-3
curve(dbinom(y,n,x),0,1,add=T)

par(mfrow=c(1,2))
y<-8; n<-10
curve(dbinom(y,n,x),0,1)
y<-3
curve(dbinom(y,n,x),0,1)


x<-8; n<-10

theta<- seq(0,1,len=100)
like <- dbinom(8,10,theta)
   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')


# 2. tapa, jossa täytyy tietää SUE


like<- function(theta){
   dbinom(x,n,theta)/dbinom(x,n,x/n)
  }
plot(like,0,1,xlab=expression(theta),
     ylab='Likelihood')

title(expression('Uskottavuusfunktio'))

#################################
# H4.3
# Count Rumford
t<-c(4,5,7,12,14,16,20,24,28,31,34,37.5,41)		
F<-c(126,125,123,120,119,118,116,115,114,113,112,111,110)		
plot(t,F)

th<- 1
curve(60+70*exp(-th*x),3,42)
#############################
# H4.5
# (a)
x<-1
theta<- seq(0,8,by=0.1)
like <- dpois(x,theta)
   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

####################################
#curve(dpois(y,x),0,8)
#Esimerkki###################
theta<- seq(0,8,by=0.1)
like <- dpois(y,theta)
   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Likelihood')

############################################
# also possible to use formula plotting, but
# we need to know the MLE in advance

#like<- function(theta){
#   dbinom(x,n,theta)/dbinom(x,n,x/n)
#  }
#plot(like,0,1,xlab=expression(theta),
#     ylab='Likelihood')

title(expression('Likelihood function'))
############################################

# (b)
x<-3
theta<- seq(0,8,by=0.1)
like <- dpois(x,theta)

   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

# (c)

theta<- seq(0,8,by=0.1)
like<-theta^(10)*exp(-10*theta)

   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

########################
lines(theta,like,lwd=.3)
#######################

# Samaan kuvioon
################
x<-1
theta<- seq(0,8,by=0.1)
like <- dpois(x,theta)
   like <- like/max(like)

plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')
x<-3
like <- dpois(x,theta)
	like <- like/max(like)
	lines(theta,like,lwd=.3)
like<-theta^(10)*exp(-10*theta)
	like <- like/max(like)
	lines(theta,like,lwd=.3)

###############################

# H4.2
# (b)
theta<- seq(0,1,by=0.01)
like <- theta^(110)*(1-theta)^(90)
	like <- like/max(like)
plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

title(expression('Tehtävä 4.2(b)'))


###############################
# H4.3
# (b)
theta<- seq(-1/6,1/12,by=0.001)
like <- (1/6+theta)^(54)*(1/6-2*theta)^(46)
	like <- like/max(like)
plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

title(expression('Tehtävä 4.3(b)'))


###############################

# H4.5

N<- seq(137,150,by=0.01)
like <- 1/N^8
	like <- like/max(like)
plot(N,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

title(expression('Tehtävä 4.5'))

##############################
# H4.6
# (c)
theta<- seq(0,0.4,by=0.001)
like <- (1-theta)^(580)*(1-(1-theta)^(20))^(91)
	like <- like/max(like)
plot(theta,like,type='l',xlab=expression(theta),
     ylab='Uskottavuus')

title(expression('Tehtävä 4.6(c)'))


###############################